Complex or imaginary shapes?

Algebra Level 5

The absolute value of a complex number would be the distance between the complex number to the origin (0,0)(0,0) in the complex plane. Or in other words, a+bi=a2+b2|a+bi| =\sqrt { { a }^{ 2 }+{ b }^{ 2 } } So for z=n|z|=n, the possible values of zz would form a circle of radius nn centered at the origin on the complex plane.

Now, supposing I create a new function: a+bi=a+b\ddagger a+bi\ddagger =|a|+|b|

And all of the possible values of zz in z=2015\ddagger z\ddagger =2015 forms a shape of area AA on the complex plane.

Find A\left\lfloor A \right\rfloor


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